scholarly journals Interaction of linear modulated waves and unsteady dispersive hydrodynamic states with application to shallow water waves

2019 ◽  
Vol 875 ◽  
pp. 1145-1174 ◽  
Author(s):  
T. Congy ◽  
G. A. El ◽  
M. A. Hoefer
Keyword(s):  
Water Waves ◽  
Large Scale ◽  
Kdv Equation ◽  
Linear Wave ◽  
Mean Flow ◽  
Undular Bore ◽  
Expansion Waves ◽  
Flow Interaction ◽  
The Kdv Equation ◽  
New Type

A new type of wave–mean flow interaction is identified and studied in which a small-amplitude, linear, dispersive modulated wave propagates through an evolving, nonlinear, large-scale fluid state such as an expansion (rarefaction) wave or a dispersive shock wave (undular bore). The Korteweg–de Vries (KdV) equation is considered as a prototypical example of dynamic wavepacket–mean flow interaction. Modulation equations are derived for the coupling between linear wave modulations and a nonlinear mean flow. These equations admit a particular class of solutions that describe the transmission or trapping of a linear wavepacket by an unsteady hydrodynamic state. Two adiabatic invariants of motion are identified that determine the transmission, trapping conditions and show that wavepackets incident upon smooth expansion waves or compressive, rapidly oscillating dispersive shock waves exhibit so-called hydrodynamic reciprocity recently described in Maiden et al. (Phys. Rev. Lett., vol. 120, 2018, 144101) in the context of hydrodynamic soliton tunnelling. The modulation theory results are in excellent agreement with direct numerical simulations of full KdV dynamics. The integrability of the KdV equation is not invoked so these results can be extended to other nonlinear dispersive fluid mechanic models.

Integrable Models

2018 ◽  
Author(s):  
S. G. Rajeev
Keyword(s):  
Fluid Mechanics ◽  
Soliton Solution ◽  
Water Waves ◽  
Heisenberg Model ◽  
Kdv Equation ◽  
Short Review ◽  
Lax Pair ◽  
Shallow Water Waves ◽  
Initial Value ◽  
The Kdv Equation

Some exceptional situations in fluid mechanics can be modeled by equations that are analytically solvable. The most famous example is the Korteweg–de Vries (KdV) equation for shallow water waves in a channel. The exact soliton solution of this equation is derived. The Lax pair formalism for solving the general initial value problem is outlined. Two hamiltonian formalisms for the KdV equation (Fadeev–Zakharov and Magri) are explained. Then a short review of the geometry of curves (Frenet–Serret equations) is given. They are used to derive a remarkably simple equation for the propagation of a kink along a vortex filament. This equation of Hasimoto has surprising connections to the nonlinear Schrödinger equation and to the Heisenberg model of ferromagnetism. An exact soliton solution is found.

scholarly journals Wave Breaking in Undular Bores with Shear Flows

Water Waves ◽  
2021 ◽  
Author(s):  
Maria Bjørnestad ◽  
Henrik Kalisch ◽  
Malek Abid ◽  
Christian Kharif ◽  
Mats Brun
Keyword(s):  
Turbulent Flows ◽  
Wave Breaking ◽  
Vries Equation ◽  
Kdv Equation ◽  
Flow Depth ◽  
Present Contribution ◽  
Undular Bore ◽  
Wave Flume ◽  
The Kdv Equation ◽  
The Waves

AbstractIt is well known that weak hydraulic jumps and bores develop a growing number of surface oscillations behind the bore front. Defining the bore strength as the ratio of the head of the undular bore to the undisturbed depth, it was found in the classic work of Favre (Ondes de Translation. Dunod, Paris, 1935) that the regime of laminar flow is demarcated from the regime of partially turbulent flows by a sharply defined value 0.281. This critical bore strength is characterized by the eventual breaking of the leading wave of the bore front. Compared to the flow depth in the wave flume, the waves developing behind the bore front are long and of small amplitude, and it can be shown that the situation can be described approximately using the well known Kortweg–de Vries equation. In the present contribution, it is shown that if a shear flow is incorporated into the KdV equation, and a kinematic breaking criterion is used to test whether the waves are spilling, then the critical bore strength can be found theoretically within an error of less than ten percent.

scholarly journals NUMERICAL STUDY OF ROSENAU-KDV EQUATION USING FINITE ELEMENT METHOD BASED ON COLLOCATION APPROACH

2017 ◽  
Vol 22 (3) ◽  
pp. 373-388 ◽  
Author(s):  
Turgut Ak ◽  
Sharanjeet Dhawan ◽  
S. Battal Gazi Karakoc ◽  
Samir K. Bhowmik ◽  
Kamal R. Raslan
Keyword(s):  
Solitary Waves ◽  
Water Waves ◽  
Numerical Study ◽  
Kdv Equation ◽  
Uniform Mesh ◽  
Shallow Water Waves ◽  
Undular Bore ◽  
Von Neumann ◽  
B Spline ◽  
Collocation Approach

In the present paper, a numerical method is proposed for the numerical solution of Rosenau-KdV equation with appropriate initial and boundary conditions by using collocation method with septic B-spline functions on the uniform mesh points. The method is shown to be unconditionally stable using von-Neumann technique. To check accuracy of the error norms L2 and L∞ are computed. Interaction of two and three solitary waves are used to discuss the effect of the behavior of the solitary waves during the interaction. Furthermore, evolution of solitons is illustrated by undular bore initial condition. These results show that the technique introduced here is suitable to investigate behaviors of shallow water waves.

scholarly journals Non-linear water waves (KdV) equation by Painlevé property and Schwarzian derivative

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3627-3641
Author(s):  
Miodrag Mateljevic ◽  
Attia Mostafa
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Water Waves ◽  
Schwarzian Derivative ◽  
Kdv Equation ◽  
Nonlinear Partial Differential Equation ◽  
New Exact Solutions ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
Linear Water Waves

The Korteweg-de Vries (KdV) equation, a nonlinear partial differential equation which describes the motion of water waves, has been of interest since John Scott Russell (1834) [4]. In present work we study this kind of equation and through our study we found that the KdV equation passes Painleve?s test, but we could not locate the solution directly, so we used Schwarzian derivative technique. Therefore, we were able to find two new exact solutions to the KdV equation. Also, we used the numerical method of Modified Zabusky-Kruskal to describe the behavior of these solutions.

Interaction of Fully-Nonlinear Internal Solitary Waves of Opposite Polarity

2021 ◽  
Author(s):  
Kevin Lamb
Keyword(s):  
Wave Field ◽  
Solitary Waves ◽  
Kdv Equation ◽  
Nonlinear Coefficient ◽  
Opposite Polarity ◽  
Internal Solitary Waves ◽  
Linear Wave ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
The Kdv Equation

<p>Previous studies have suggested that fully nonlinear internal solitary waves (ISWs) are very soliton-like as the interaction of two ISWs results in only very small changes in amplitude of the interacting ISWs and in the production of a very small amplitude wave train. Previous studies have, however, considered ISWs with the polarity predicted by the sign of the quadratic nonlinear coefficient of the KdV equation. The Gardner equation, which is an extension of the KdV equation that includes a cubic nonlinear term, has ISWs of two polarities (i.e., waves of depression and elevation) when the cubic coefficient of the Gardner equation is positive. These waves are soliton solutions of the Gardner equations.  In this talk I will discuss the interaction of ISWs of opposite polarity in continuous asymmetric three layer stratifications. Regions in parameter space where ISWs of opposite polarity exist will be discussed and I will demonstrate via fully nonlinear numerical simulations that the interaction of ISWs of opposite polarity waves are far from soliton-like: their interaction can result in very large changes in wave amplitude and may produce a very complicated wave field with multiple large ISWs, a large linear wave field and breather-like waves.<span> </span></p>

The Shape, Propagation and Mean-Flow Interaction of Large-Scale Weather Systems

1983 ◽  
Vol 40 (7) ◽  
pp. 1595-1612 ◽  
Author(s):  
Brian J. Hoskins ◽  
Ian N. James ◽  
Glenn H. White
Keyword(s):  
Large Scale ◽  
Mean Flow ◽  
Flow Interaction

Shallow-water waves, the Korteweg-deVries equation and solitons

1971 ◽  
Vol 47 (4) ◽  
pp. 811-824 ◽  
Author(s):  
N. J. Zabusky ◽  
C. J. Galvin
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Water Tank ◽  
Shallow Water Waves ◽  
Downstream Location ◽  
Low Dissipation ◽  
The Kdv Equation ◽  
Upstream Station

A comparison of laboratory experiments in a shallow-water tank driven by an oscillating piston and numerical solutions of the Korteweg-de Vries (KdV) equation show that the latter can accurately describe slightly dissipative wavepropagation for Ursell numbers (h1L2/h03) up to 800. This is an input-output experiment, where the initial condition for the KdV equation is obtained from upstream (station 1) data. At a downstream location, the number of crests and troughs and their phases (or relative locations within a period) agree quantitatively with numerical solutions. The crest-to-trough amplitudes disagree somewhat, as they are more sensitive to dissipative forces. This work firmly establishes the soliton concept as necessary for treating the propagation of shallow-water waves of moderate amplitude in a low-dissipation environment.

scholarly journals Dynamic soliton–mean flow interaction with non-convex flux

2021 ◽  
Vol 928 ◽  
Author(s):  
Kiera van der Sande ◽  
Gennady A. El ◽  
Mark A. Hoefer
Keyword(s):  
Wave Propagation ◽  
Large Scale ◽  
Fluid Dynamic ◽  
Internal Gravity Wave ◽  
Mkdv Equation ◽  
Stratified Fluids ◽  
Mathematical Framework ◽  
Mean Flow ◽  
Wave Trapping ◽  
Flow Interaction

The interaction of localised solitary waves with large-scale, time-varying dispersive mean flows subject to non-convex flux is studied in the framework of the modified Korteweg–de Vries (mKdV) equation, a canonical model for internal gravity wave propagation and potential vorticity fronts in stratified fluids. The effect of large amplitude, dynamically evolving mean flows on the propagation of localised waves – essentially ‘soliton steering’ by the mean flow – is considered. A recent theoretical and experimental study of this new type of dynamic soliton–mean flow interaction for convex flux has revealed two scenarios where the soliton either transmits through the varying mean flow or remains trapped inside it. In this paper, it is demonstrated that the presence of a non-convex cubic hydrodynamic flux introduces significant modifications to the scenarios for transmission and trapping. A reduced set of Whitham modulation equations is used to formulate a general mathematical framework for soliton–mean flow interaction with non-convex flux. Solitary wave trapping is stated in terms of crossing modulation characteristics. Non-convexity and positive dispersion – common for stratified fluids – imply the existence of localised, sharp transition fronts (kinks). Kinks play dual roles as a mean flow and a wave, imparting polarity reversal to solitons and dispersive mean flows, respectively. Numerical simulations of the mKdV equation agree with modulation theory predictions. The mathematical framework developed is general, not restricted to completely integrable equations like mKdV, enabling application beyond the mKdV setting to other fluid dynamic contexts subject to non-convex flux such as strongly nonlinear internal wave propagation that is prevalent in the ocean.

scholarly journals Nonlinear water waves (KdV) equation and Painleve technique

2015 ◽  
Vol 4 (2) ◽  
pp. 216
Author(s):  
Attia Mostafa
Keyword(s):  
Exact Solutions ◽  
Water Waves ◽  
Kdv Equation ◽  
Nonlinear Pde ◽  
Nonlinear Water Waves ◽  
Third Order ◽  
The Third ◽  
Painlevé Property ◽  
The Kdv Equation ◽  
De Vries

<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

scholarly journals The Korteweg-de Vries equation and water waves. Part 2. Comparison with experiments

1974 ◽  
Vol 65 (2) ◽  
pp. 289-314 ◽  
Author(s):  
Joseph L. Hammack ◽  
Harvey Segur
Keyword(s):  
Initial Data ◽  
Water Waves ◽  
Entire Range ◽  
Vries Equation ◽  
Kdv Equation ◽  
Asymptotic Shape ◽  
Wide Range ◽  
The Kdv Equation ◽  
De Vries ◽  
Wave Forms

The Korteweg-de Vries (KdV) equation is tested experimentally as a model for moderate amplitude waves propagating in one direction in relatively shallow water of uniform depth. For a wide range of initial data, comparisons are made between the asymptotic wave forms observed and those predicted by the theory in terms of the number of solitons that evolve, the amplitude of the leading soliton, the asymptotic shape of the wave and other qualitative features. The KdV equation is found to predict accurately the number of evolving solitons and their shapes for initial data whose asymptotic characteristics develop in the test section of the wave tank. The accuracy of the leading-soliton amplitudes computed by the KdV equation could not be conclusively tested owing to the viscous decay of the measured wave amplitudes; however, a procedure is presented for estimating the decay in amplitude of the leading wave. Computations suggest that the KdV equation predicts the amplitude of the leading soliton to within the expected error due to viscosity (12%) when the non-decayed amplitudes are less than about a quarter of the water depth. Indeed, agreement to within about 20% is observed over the entire range of experiments examined, including those with initial data for which the non-decayed amplitudes of the leading soliton exceed half the fluid depth.

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